### Abstract

Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are considered. If the family of expectations of such a process is a spanning subset of the space, it is proved that the scaling family of operators of the process under consideration is a uniquely determined multiplicative group of operators. If the expectation-function of the process is continuous, it is proved that the expectations of the process have power-growth with exponent greater than or equal to 0, that is, their norm is less than a nonnegative constant times such apower-function, provided that the linear space spanned by the expectations has category 2 (in the sense of Baire) in its closure. It is shown that OSS processes whose expectation-function is differentiable on an interval (s_{0}, ∞), for some s_{0} ≥ 1, have a unique scaling family of operators of the form {s^{H}: s > 0}, if the expectations of the process span a dense linear subspace of category 2. The existence of a scaling family of the form {s^{H}: s > 0} is proved for proper Hilbert space OSS processes with an Abelian scaling family of positive operators.

Original language | English (US) |
---|---|

Article number | 82838 |

Journal | Journal of Applied Mathematics and Stochastic Analysis |

Volume | 2006 |

DOIs | |

State | Published - 2006 |

### ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics